#6596846

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573549
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3264	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,23)(16,22)(17,19)(18,21)(20,24)(25,28)(26,27);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(20,27)(21,22)(23,28);(7,12)(9,10)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,23)(22,28);(5,7)(6,10)(8,12)(9,11)(13,16,17,18)(14,21,19,22)(15,24,25,26)(20,23,27,28);(7,12)(9,10)

Polycyclic presentation:

2^9|b^2=fhi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283965, #72284363, #72284532
maximal normal subgroup of	#72283965, #72284363, #72284532

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504042);